F. Blanchet-Sadri and F.D. Gaddis, "On a Product of Finite Monoids." Semigroup Forum, Vol. 57, 1998, pp DOI: 10.

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1 On a Product of Finite Monoids By: F. Blanchet-Sadri and F. Dale Gaddis F. Blanchet-Sadri and F.D. Gaddis, "On a Product of Finite Monoids." Semigroup Forum, Vol. 57, 1998, pp DOI: /PL Made available courtesy of Springer Verlag: ***The original publication is available at ***Reprinted with permission. No further reproduction is authorized without written permission from Springer Verlag. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document.*** Abstract: In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S 1,, S m, a product m (S m,, S 1, S 0 ). We give a representation of the free objects in the pseudovariety m (W m,, W 1, W 0 ) generated by these (m + 1)-ary products where S i W i for all 0 i m. We then give, in particular, a criterion to determine when an identity holds in m (J 1,,J 1,J 1 ) with the help of a version of the Ehrenfeucht-Fraïssé game (J 1 denotes the pseudovariety of all semilattice monoids). The union (J 1,, J 1, J 1 ) turns out to be the second level of the Straubing s dot-depth hierarchy of aperiodic monoids. Article: 1. Introduction The theory of varieties of Eilenberg constitutes an elegant framework for discussing relationships between combinatorial descriptions of languages and algebraic properties of their recognizers. The interplay between the two points of view leads to interesting classifications of languages and finite monoids. A variety of languages is often described as the smallest variety closed under a given class of operations, such as boolean operations, concatenation product, star, and so on. A variety of finite monoids or semigroups (or pseudovariety) is also often described with the help of operations: join, semidirect and two-sided semidirect products, and Sch utzenberger product, to name a few. In view of Eilenberg s correspondence between varieties of languages and varieties of finite monoids or semi-groups, we may expect some relationships between the operations on languages (of combinatorial nature) and the operations on monoids or semigroups (of algebraic nature). Examples of correspondence between varieties of languages and varieties of finite monoids or semigroups include: the variety of rational (or regular) languages corresponds to the variety of all finite monoids [19]; the aperiodic or star-free languages correspond to the finite aperiodic monoids [28]; the piecewise testable languages to the finite I-trivial monoids [29] and the locally testable languages to the finite locally idempotent and commutative semigroups [14, 20]. In this paper, we construct an (m + 1)-ary product of finite monoids and give a relationship between this operation on monoids and a version of the EhrenfeuchtFraïssé game corresponding to the levels of the so-called dot-depth hierarchy of aperiodic languages The Straubing hierarchy A monoid S is said to be aperiodic if all groups in S are trivial. The pseudovariety of all aperiodic monoids is denoted by A. A result of Schützenberger [28] enables us to describe the *-variety A of aperiodic languages

2 corresponding to the pseudovariety A. For a finite alphabet A, the class A * A is the least class of languages of A * (the free monoid generated by A) satisfying the following three conditions: A* A is closed under finite boolean operations, If L, L' A* A, then the concatenation LL' A* A, {u} A * A for all u A *. Cohen and Brzozowski [15] introduced the dot-depth hierarchy for the aperiodic languages of A + = A * \ {1} (1 denotes the empty word), and Straubing [31] defined another hierarchy for the aperiodic languages of A *. The Straubing hierarchy is a doubly indexed hierarchy for the class A * A. It grows out in a natural manner from the applications of concatenation and boolean operations. We proceed inductively starting with A * V 0 = {, A * }. Assuming that A*V k - 1 is already defined for some k > 0, the class A * V k is defined as the boolean closure of all languages of the form L 0 a 1 L 1 a i L i, with L 0,, L i A * V k-1 and a 1,, a i A. Clearly, A * V 0 A * V 1 and the union is the class A * A of all aperiodic languages of A *. Within each A * V k (k > 0) one can again establish a hierarchy by defining A * V k,m to be the boolean closure of the languages L 0 a 1 L 1 a i L i with i m and with L 0,, L i A * V k-1 and a 1,, a i A as before. Then A * V k,1 A * V k,2 and the union is A * V k. The class V k = {A * V k } is a *-variety of languages for k 0, and the classes V k,m = {A * V k,m } with k > 0, m > 0 form *-varieties of languages. For each of the *-varieties V k (k 0) we will denote by V k the corresponding pseudovariety of monoids. Similarly, V k,m will denote the pseudovariety of monoids corresponding to V k,m for k > 0, m > 0. We have the following result of Simon [29]: V 1 is decidable or V 1 = J. In the language of the Green relations, J is the pseudovariety of monoids in which the I-relation is trivial Games and the Straubing hierarchy Thomas [35] gave a purely logical proof of the strictness of the Straubing hierarchy (the original proof of the strictness of the dot-depth hierarchy is due to Brzozowski and Knast [13]). Thomas proof is based on an Ehrenfeucht-Fraïssé game which we now describe. Let A be a finite alphabet. Consider the set of symbols L = {<} {Qa a A}. We identify each word u of A * (of length u ) with an L-structure where u = {1,, u } represents the set of positions of letters in u, < u is the natural ordering on the integers 1,, u and, for each a A, is the subset of u defined by i if and only if the ith letter of u is an a. Let u = ( u, < u, ( ) a A ), v = ( v, < v, ( ) a A ) be L-structures. Let = (m 1,,m k ) be a k-tuple of positive integers, where k 0. The Ehrenfeucht-Fraïssé game (u, v) corresponding to and u, v is played by two players, I and II, according to the following rules: The letter k is the number of moves each player has to make in the course of a play of the game (u,v). These moves are begun by Player I, and both players move alternately. The ith move consists of choosing m i positions from u or from v. If Player I chooses m i positions from u in his ith move, then Player II must choose m i positions from v in his ith move. If Player I chooses m i positions from v in his ith move, then Player II must choose m i positions from u. After the kth move of Player II the play is completed. Altogether some positions p 1,,p n u and q 1,,q n v have been chosen where n = m m k. Player II has won the play if the following two conditions are satisfied: p i < u p j if and only if q i < v q j for all 1 i,j n, pi if and only if q i for all 1 < i < n and a A.

3 We say that Player II has a winning strategy in (u,v) and write v if it is possible for him to win each play. The equivalence naturally defines a congruence on A * of finite index. The -class of u is {v A * v} and will be denoted by. The set of all -classes, A*/, will be denoted by A * /. This set becomes a monoid by considering the operation [ = ; acts as unit. The importance of lies in the fact that V k can be described in terms of the congruences. Thomas [34, 35], and Perrin and Pin [21] infer that the monoids A * /(m 1,...,m k ) (respectively A * /(m, m 1,,m k-1 )) form a family of finite monoids that generate V k (respectively V k,m ) in the sense that every finite aperiodic monoid in V k (respectively V k,m ) is a morphic image of a monoid of the form A * /(m 1,,m k ) (respectively A * /(m, m 1,, m k- 1)). In [12], we give a reduced family of generators for V k. In particular, we show that the monoids A * /(m,1) form a family of monoids that generate V 2. The problem remains open as to whether V k is decidable for k 2. Partial results have been obtained mostly for the 2nd level (Blanchet-Sadri [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Cowan [16], Pin [25], Straubing [25, 32, 33] and Weil [33, 36, 38]). For fixed, we define the pseudovariety of monoids, as follows: an A-generated monoid S is in if and only if S is a morphic image of A * /. Note that V (1) = J 1 the pseudovariety of semilattice monoids. The congruences can be defined inductively as follows: First, u v if and only if α(u) = α(v), or u and v have the same set of letters. Then, u v if and only if u and v have the same set of subwords of length m. Now, if u = a 1 a n is a word on A, then denote the segment a i a j by u[i, j] for all 1 i j n. Lemma 1.[Blanchet-Sadri [3]] Let u and v be words on a finite alphabet A. For all positive integers m and tuples of positive integers, we have that u v if and only if For every p 1,..., p m u (where p 1 < < p m ), there exist q 1,...,q m v (where q 1 < < q m ) such that 1. p i if and only if q i for all 1 i m and a A, 2. u[1, p 1 1] v[1, q 1 1], 3. u[p i + 1, p i+1 1] v[q i + 1, q i+1 1] for all 1 i < m, 4. u[p m + 1, u ] v[q m + 1, v ], and For every q 1,, q m v (where q 1 < < q m ), there exist p1,,p m u (where p 1 < < p m ) such that 1 4 hold. When there is no chance of confusion, we will write u instead of ab ba in {a, b} *.. We note that ab = ba in {a, b} * /(1), but Definition 2. Let u be a word on a finite alphabet A. For all positive integers m and tuples of positive integers, (u) consists of the set For example, if A = {a, b}, then a (2,1) (abc) is the set

4 Lemma 1 states that u v if and only if (u) = (v). In Section 2 of this paper, some background information is presented. In Section 3, we first review the twosided semidirect product of monoids. Next, we associate with a monoid S 0 and m commutative monoids S 1,,S m, an (m + 1)-ary product m (S m,,s 1, S 0 ) (Proposition 5). The pseudovariety m (W m,...,w 1, W 0 ) is defined as being generated by these (m + 1)-ary products where S i W i for all 0 i m. Section 3.1 gives a criterion to determine when an identity holds in the two-sided semidirect product J 1 ** with the help of, and Section 3.2 gives such a criterion for m (J 1,,J 1, ) with the help of. The essential ingredients in our proofs are a two-sided semidirect product representation of the free objects in V ** W due to Almeida and Weil [2] stated in Theorem 3, and our representation of the free objects in m (W m,,w 1, W 0 ) stated in Theorem 6. The equalities J 1 ** =, m (J 1,,J 1, ) = and V 2 = (J 1,,J 1,J 1 ) result (Corollaries 4, 7 and 8). 2. Preliminaries This section is devoted to reviewing basic properties of pseudovarieties of monoids and of *-varieties of recognizable languages. The reader is referred to the books of Almeida [1], Eilenberg [17] and Pin [22] for further definitions and background Pseudovarieties of monoids A pseudovariety (of monoids) V is a family of finite monoids that satisfies the following two conditions: If S V and T < S (T divides S), then T V, If S, T V, then the cartesian product S T V. For any family C of finite monoids, we denote by (C) the least pseudovariety of monoids containing C. Clearly, S (C) if and only if S < S 1 S i with S 1,..., S i C. We call (C) the pseudovariety of monoids generated by C. An (monoid) identity on an alphabet A is a pair (u, v) of words of A *, usually indicated by a formal equality u = v. Given u, v A * and given a monoid S, we will say that S satisfies the identity u = v (or that the identity u = v holds in S) and we write S = u = v if φ(u) = φ(v) for every morphism φ: A * S of monoids. For an identity u = v and a pseudovariety V, the notation V = u = v will abbreviate the fact that each S V satisfies u = v. Work of Eilenberg and Schützenberger [18] showed that pseudovarieties of monoids are ultimately defined by sequences of identities (that is, a monoid belongs to the given pseudovariety if and only if it satisfies all but finitely many of the identities in the sequence), and that finitely generated pseudovarieties of monoids are defined by sequences of identities or are equational (that is, a monoid belongs to the given pseudovariety if and only if it satisfies all the identities in the sequence). The free object on the alphabet A in the variety generated by a pseudovariety V will be denoted by F A (V). We say that F A (V) has the universal property for V on A in the following sense: for any S V and any function φ : A S, there exists (precisely) one morphism ψ : F A (V) S such that the diagram

5 commutes, where A F A (V) is the function that sends each a A to a (we also say that the map A F A (V) has the universal property). The morphism ψ is defined by ψ(a 1 a n ) = φ(a 1 ) φ(a n ) and is said to be the natural extension of φ to F A (V) *-Varieties of languages Let A be a finite alphabet. Let L be a language of A *. We define a congruence ~ L on A * as follows: u ~ L v holds if xuy L if and only if xvy L for all x, y A *. The congruence ~ L is called the syntactic congruence of L, and the quotient monoid A * / ~ L, which we denote by S(L), is called the syntactic monoid of L. The language L is recognizable if and only if S(L) is a finite monoid. A *-class V = {A * V} consists of a family A * V of recognizable languages of A * defined for every finite alphabet A. We will say that a *-class V satisfying the following three conditions is a *-variety (of languages): A * V is closed under boolean operations, If L A * V and a A, then the sets a -1 L = {u E A * au L} and La -1 = {u A * ua L} are in A * V, If φ : B * A * is a morphism of monoids and if L A * V, then φ -1 (L) B * V. Eilenberg [17] established a one-to-one correspondence between pseudovarieties of monoids and *-varieties of languages. For each *-variety V, define the pseudovariety generated by the syntactic monoids of languages in V. For each pseudovariety of monoids V, define the *- variety V by 3. Products of monoids We now proceed with the products mentioned in the introduction. A related product is the Schützenberger product that has been studied by several authors [23, 24, 25, 26, 28, 30] The two-sided semidirect product In this section, we review the definition of the two-sided semidirect product of monoids, a 2-ary product introduced by Rhodes and Tilson [27]. Let S be an additively written monoid with unit 0 (commutativity for S is not assumed). Let T be a multiplicatively written monoid with unit 1. A left unitary action of T on S is a function where for all x 1, x 2 S and y 1, y 2 T, f satisfies the following four conditions: y 1 (x 1 + x 2 ) = y 1 x 1 + y 1 x 2, y 1 (y 2 x 1 ) = (y 1 y 2 )x 1, 1x 1 = x 1, y 1 0 = 0.

6 Let f be a left unitary action of T on S. Then associated with S T and f is the semidirect product S * T with operation This operation is associative and S * T is a monoid with unit (0, 1). A right unitary action of T on S is a function where for all x 1, x 2 S and y 1, y 2 T, g satisfies the following four conditions: (x 1 + x 2 )y 1 = x 1 y 1 + x 2 y 1, (x 1 y 1 )y 2 = x 1 (y 1 y 2 ), x 1 1 = x 1, 0y 1 = 0. Let f be a left unitary action and g be a right unitary action of T on S. Assume for all x S and y 1, y 2 T, that f and g satisfy the following condition: y 1 (xy 2 ) = (y 1 x)y 2. Then associated with S T and f, g is the two-sided semidirect product S ** T with operation This operation is associative and S ** T is a monoid with unit (0, 1). When g is trivial, then S ** T is in fact a semidirect product. Neither * nor ** is associative on monoids. For pseudovarieties of monoids V and W, their semidirect product V * W (respectively two-sided semidirect product V ** W) is defined to be the pseudovariety of monoids generated by all semidirect products S * T (respectively two-sided semidirect products S ** T) with S V and T W. The operation * is associative on pseudovarieties of monoids but ** is not. We now give the following representation of free objects for V ** W obtained by Almeida and Weil. In general, the free object on the alphabet A in the variety generated by a pseudovariety V, F A (V), does not lie in V. We have F A (V) E V if and only if F A (V) is finite. Theorem 3. [Almeida and Weil [2]] Let V and W be two pseudovarieties of monoids such that F A (V) and F A (W) are finite for all finite alphabets A. Then so is V ** W. Moreover, if A is a finite alphabet, there exists a one-to-one morphism from F A (V ** W) into F B (V) ** F A (W) given by a ((1, a, 1), a), where B = F A (W) A F A (W), the left unitary action of F A (W) on F B (V) is defined by x(y, a, z) = (xy, a, z) and the right unitary action by (y, a, z)x = (y, a, zx) for all x, y, z F A (W) and a A. We end this section with a criterion to determine when an identity is satisfied in the two-sided semidirect product J1 **.

7 Corollary 4. Let A be a finite alphabet and be a tuple of positive integers. An identity u = v on A holds in J 1 ** if and only if u v. Consequently, an A-generated monoid S belongs to J 1 ** if and only if S is a morphic image of A*/(1, ), or J 1 ** =. Proof. Let T be the multiplicatively written monoid F A ( ) with unit 1 (F A ( is isomorphic to A * / ) and let S be the additively written monoid F T A T (J 1 ) with unit 0. Consider the left unitary action of T on S defined by x(y, a, z) = (xy, a, z) and the right unitary action of T on S defined by (y, a, z)x = (y, a, zx) for all x, y, z T and a A, and the associated two-sided semidirect product S ** T. Now consider the one-to-one morphism of Theorem 3, φ : F A (J 1 ** ) S ** T, where for all a A, φ(a) = ((1, a, 1), a). If u and v are words on A, then u = v holds in J 1 ** if and only if u = v holds in F A (J 1 ** ) if and only if φ(u) = φ(v) if and only if u v. To see that φ(u) = φ(v) and u v are equivalent, the morphism φ maps the word u = a 1 a i into the 2-tuple and v = b 1...b j into The equality φ(u) = φ(v) holds if and only if corresponding components of the 2- tuples (1) and (2) are equal. If u' (respectively v') denotes the first component of (1) (respectively (2)), then the condition the first components of (1) and (2) are equal is equivalent to S = u' = v' or (u) = (v), and the condition the second components of (1) and (2) are equal is equivalent to T = u = v or u v. The condition (u) = ~(1, m) (v) (which is clearly equivalent to ~(1, m) (u) = ~(1, m) (v) by Definition 2) implies the condition u m v. Hence, we conclude that W(u) = W(v) if and only if u (1, m) v. The equality J 1 ** V k = V k+1,1 is known to Weil (this is a particular case of Proposition 2.12 in [371) An (m + 1)-ary product In this section, we construct an (m + 1)-ary product of monoids for each positive integer m. Let S 0 be a multiplicatively written monoid with unit 1. Let S i (1 i m) be an additively written commutative monoid with operation + i and unit 0 i. For the following discussion, we will use subscripts to indicate the monoid and different letters to indicate different members within the monoid (for instance x i, y i Si). For 0 i, j m and i + j m, let h i,j be a function Now assume that h 0,0 (x 0, y 0 ) is the product x 0 y 0 in S 0 and that 1. x i (x j + j y j ) = x i x j + i + j x i y j for all 0 i m, 1 j m with i + j m, 2. (x i + i y i )x j = x i x j + i+j y i x j for all 1 i m, 0 j m with i + j m, 3. x i (x j x k ) = (x i x j )x k for all 0 i, j, k m with i + j + k m, 4. x i 1 = x i = 1x i for all 0 < i < m, 5. x i 0 j = 0 i+j for all 0 i m, 1 j m with i + j m, 6. 0 i x j = 0 i+j for all 1 i m, 0 j m with i + j m.

8 Then associated with S m S 1 S 0 and the functions h i,j is the (m + 1)-ary product m (S m,..., S 1, S 0 ) with operation where z 0 = x 0 y 0 and for all 1 < i < m, Proposition 5. The above operation is associative and m (S m,, S 1, S 0 ) is a monoid with unit (0 m,...,0 1,1). Proof. Let x = (x m,, x 1, x 0 ), y = (y m,, y 1, y 0 ), z = (z m,, z 1, z 0 ) S m S 1 S 0. Then for all 1 i m, we have ((xy)z) i = (x i y 0 + i + i x 0 y i )z 0 + i (x i-1 y 0 + i-1 + i-1 x 0 y i-1 )z 1 + i + i (x 1 y x 0 y 1 )z i-1 (+ i (x 0 y 0 )z i, = ((x i y 0 )z 0 + i + i (x 0 y i )z 0 ) + i ((x i-1 y 0 )z 1 + i + i (x 0 y i-1 )z 1 ) + i + i ((x 1 y 0 )z i-1 + i (x 0 y 1 )z i-1 ) + i (x 0 y 0 )z i (Condition 2), = (x i y 0 )z 0 + i + i (x 0 y i )z 0 + i (x i-1 y 0 )z 1 + i + i (x 0 y i-1 )z 1 + i + i (x 1 y 0 )z i-1 + i (x 0 y 1 )z i-1 + i (x 0 y 0 )z i (Associativity in S i ), = x i (y 0 z 0 ) + i + i x 0 (y i z 0 ) + i x i-1 (y 0 z 1 ) + i + i x 0 (y i-1 z 1 ) + i + i x 1 (y 0 z i-1 ) + i x 0 (y 1 z i-1 ) + i x 0 (y 0 z i ) (Condition 3), = x i (y 0 z 0 ) + i x i-1 (y 1 z 0 ) + i x i-1 (y 0 z 1 ) + i + i x 1 (y i-1 z 0 ) + i + i x 1 (y 0 z i-1 )+ i x 0 (y i z 0 )+ i + i x 0 (y 0 z i ) (Commutativity in S i ), = x i (y 0 z 0 ) + i x i-1 (y 1 z y 0 z 1 ) + i + i x 1 (y i-1 z 0 + i-1 + i-1 y 0 z i-1 ) + i x 0 (y i z 0 + i + i y 0 z i ) (Condition 1), = (x(yz)) i. Clearly ((xy)z) 0 = (x 0 y 0 )z 0 = x 0 (y 0 z 0 ) = (x(yz)) 0. For all 1 i m, we have (x(0 m,, 0 1, 1)) i = x i 1 + i x i i + i x 0 0 i, = x i + i x i i + i x 0 0 i (Condition 4), = xi +i 0i +i +i 0i (Condition 5), = xi. Similarly, we have ((0 m,, 0 1,1)x) i = x i. Clearly, (x(0 m,, 0 1,1)) 0 = x 0 1 = x 0 = 1x 0 = ((0 m,, 0 1,1)x) 0. Therefore, (0 m,,0 1,1) acts as unit. For a pseudovariety of monoids W 0 and commutative pseudovarieties of monoids W 1,, W m, their (m + 1)-ary product m (W m,..., W 1, W 0 ) is defined to be the pseudovariety of monoids generated by all (m + 1)-ary products m (S m,, S 1, S 0 ) with S i W i for all 0 i m. We give the following representation of free objects for m (W m,, W 1, W 0 ). For a positive integer m, denote a sequence of 2m 1 1 s. For instance, (1, a, ) = (1, a, 1, 1, 1, 1, 1). will

9 Theorem 6.Let m be a positive integer. Let W 0,, W m be pseudovarieties of monoids (with W 1,, W m commutative) such that F A (W i ) is finite for all finite alphabets A (0 i m). Then so is m (W m,, W 1, W 0 ). Moreover, if A is a finite alphabet, there exists a one-to-one morphism given by a ((1, a, ),, (1, a, ), (1, a, 1), a), where B i = F A (W 0 ) A (F A (W 0 ) (A {1})) i-1 F A (W 0 ) for all 1 i m, and the functions h i,j are defined by for all x 0,..., x i, y 0,..., y j F A (W 0 ), a 1, b 1 A, and a 2,..., a i, b 2,..., b j A {1} (0 i,j m with i+j m). Proof. Let F be the submonoid of m ( (W m ),, (W 1 ), FA(W 0 )) generated by A~ = { a A} where = ((1, a, ),, (1, a, ~12), (1, a, 1), a). We show that F is isomorphic to F A ( m (W m,,w 1, W 0 )). In order to do this, we show that the mapping from A into F given by a has the appropriate universal property, or for any S m (W m,..., W 1, W 0 ) and any function φ : A S, there exists (precisely) one morphism : F S such that the diagram commutes (or = φ(a) for every a A). Let S E o m (W m,..., W 1, W 0 ) and let φ : A S be a function. Then there exist Sl W l (0 l m) and an (m + 1)-ary product m (S m,,s 1,S 0 ) such that S < m (S m,,s 1,S 0 ), say S is a morphic image of a submonoid T of m (S m,,s 1,S 0 ) under an onto morphism : T S. Let ψ : A T be such that ψ = φ. Thus, we have a mapping ψ : A m (S m,,s 1,S 0 ). The existence of a unique extension of φ to F follows from the existence of a unique extension of ψ to F, and so we proceed to work with the function ψ. Let ψ 0,...,ψ m be the components of ψ, that is the functions ψl: A S l (0 l m) such that ψ(a) = (ψ m (a),...,ψ 1 (a), ψ 0 (a)) for any a A. By the universal property of F A (W 0 ), let be the unique morphism from F A (W 0 ) to S 0 such that the diagram

10 commutes, where A F A (W 0 ) is the function that sends each a A to a. For 1 l m, let function that maps (x 0, a 1, x 1,, a l, x l ) to : B l S l be the where are those a s that are not 1 s and 1 = i 1 < < i r l (that is, the result of (x 0 ) S 0,, S 0,, S 0,, and (x l ) S 0 for any x 0,..., x l F A (W 0 ), a 1 A, and a 2,..., a l (A {1})). For instance, By the universal property of (W l ), let be the unique morphism from (W l ) to S l such that the diagram commutes, where BtFB,(Wt) is the function that sends each (x 0, a 1, x 1,..., a l, x l ) B l to (x 0, a 1, x 1,..., a l, x l ). We wish to show that there exists a unique morphism such that the diagram commutes (or = ψ(a) for any a A). The unique function that may have such a property has to satisfy the following condition:

11 Once we establish that the above formula defines a function, the proof will be complete, since it will then certainly define a morphism. Suppose that... =... in F (a 1,, a i, b 1,, b j A) (we also suppose that i,j m; the other cases are simpler). In view of the definition of the operation in the (m + 1)-ary product m ( (W m ),, (W 1 ), F A (W 0 )), the preceding equality means that we have the following equalities, respectively in (W l ) (1 l m) and F A (W 0 ): Here, sum l (a 1... a i ) is the sum + l in S l of all expressions of the form (where c 1 {a 1,..., a i } and c 2,..., c l E ({a 1,..., a i } {1})) satisfying: If,..., are those c s that are not 1 s (1 = k 1 < < k r l), all the x s are equal to 1 except possibly x 0,,, and x l. In such cases, by letting =,, =, we have x 0 = a 1, =,, =, x l = a i. For instance, for l = 8, such an expression might be (for i 16): To see that the calculations of (3) and (4) hold, we have We have = (... (( ) )...) for each 1 l i, and so ( ) m = (1, a 1, )a 2 + m (1, a 1, )(1, a 1, 1)+ m + m (1, a 1, 1)(1, a 2, + m a 1 (1, a 2, ), = (1, a 1,, 1, a 2 ) + m (1, a 1,, a 2, 1) + m + m (1, a 1, 1, a 2, ) + m (a 1, a 2, ), = sum m (a 1 a 2 ), 3 = (1, a 1, )a (1, a 1, )(1, a 2, 1) + 3 (1, a 1,1)(1, a 2, ) + 3 a 1 (1, a 2, ), = (1, a 1, 1,1,1,1, a 2 ) + 3 (1, a 1,1,1,1, a 2,1) + 3 (1, a 1, 1, a 2,1,1,1) + 3 (a 1, a 2,1,1,1,1,1), = sum 3 (a 1 a 2 ), 2 = (1, a 1, )a (1, a 1,1)(1, a 2, 1) + 2 a 1 (1, a 2, ), = (1, a 1, 1, 1, a 2 ) +2 (1, a 1, 1, a 2, 1) +2 (a 1, a 2, 1, 1, 1) = sum 2 (a 1 a 2 ), 1 = (1, a 1, 1)a a 1 (1, a 2, 1) = (1, a 1, a 2 ) +1 (a 1, a 2, 1) = sum 1 (a 1 a 2 ), 0 = a 1 a 2. Combining all the components, we see that = (sum m (a 1 a 2 ),, sum 1 (a 1 a 2 ), a 1 a 2 ).

12 Next we have m = sum m (a 1 a 2 )a 3 + m sum m-1 (a 1 a 2 )(1, a 3, 1) + m + m sum 1 (a 1 a 2 )(1, a 3, ) + m a 1 a 2 (1, a 3, ) = sum m (a 1 a 2 a 3 ), 4 = sum 4 (a 1 a 2 )a sum 3 (a 1 a 2 )(1, a 3, 1) + 4 sum 2 (a 1 a 2 )(1, a 3, ) + 4 sum 1 (a 1 a 2 )(1, a 3, ) + 4 a 1 a 2 (1, a 3, ) = sum 4 (a 1 a 2 a 3 ), 3 = sum 3 (a 1 a 2 )a sum 2 (a 1 a 2 )(1, a 3, 1) + 3 sum 1 (a 1 a 2 )(1, a 3, ) + 3 a 1 a 2 (1, a 3, ), = (1, a 1, 1, 1, 1, 1, a 2 a 3 ) + 3 (1, a 1, 1, 1, 1, a 2, a 3 ) + 3 (1, a 1, 1, a 2,1, 1, a 3 ) + 3 (a 1, a 2, 1,1,1,1, a 3 ) + 3 (1, a 1, 1, 1, a 2, a 3, 1) + 3 (1, a 1, 1, a 2, 1, a 3, 1) + 3 (a 1, a 2, 1,1,1, a 3, 1) + 3 (1, a 1, a 2, a 3, 1, 1), + 3 (a 1, a 2, 1, a 3, 1, 1, 1) + 3 (a 1 a 2, a 3, 1, 1, 1, 1,1) = sum 3 (a 1 a 2 a 3 ), 2 = sum 2 (a 1 a 2 )a sum 1 (a 1 a 2 )(1, a 3,1) + 2 a 1 a 2 (1, a 3, ), = (1, a 1, 1, 1, a 2 a 3 ) + 2 (1, a 1, 1, a 2, a 3 ) + 2 (a 1, a 2, 1, 1, a 3 ) + 2 (1, a 1, a 2, a 3, 1) + 2 (a 1, a 2, 1, a 3, 1) + 2 (a 1 a 2, a 3, 1, 1, 1) = sum 2 (a 1 a 2 a 3 ), 1 = sum 1 (a 1 a 2 )a a 1 a 2 (1, a 3, 1), = (1, a 1, a 2 a 3 ) + 1 (a 1, a 2, a 3 ) + 1 (a 1 a 2, a 3, 1) = sum 1 (a 1 a 2 a 3 ), 0 = a 1 a 2 a 3. Combining all the components, we see that a1 a2 a3 = (sum m (a1a2a3),..., sum1 (a1 a2a3), a1 a2a3). Now by a similar process we produce ( ) for each 4 l < i. Finally, we get (( ) ) m = sum m (a 1 a i-1 )a i + m sum m-1 (a 1 a i-1 )(1, ai,1) + m + m sum 1 (a 1 a i-1 )(1, a i, ) + m a 1 a i-1 (1, a i, ), = sum m (a 1 a i ), (( ) ) 1 = sum 1 (a 1 a i-1 )a i + 1 a 1 a i-1 (1, a i, 1), = (1, a 1, a 2 a i ) (a 1 a i-1, a i, 1) = sum 1 (a 1 a i ), (( ) ) 0 = a 1... a i. Combining all the components, we see that = (sum m (a 1 a i ),, sum 1 (a 1 a i ), a 1 a i ). Applying to both members of the equalities (3) and (4) respectively the morphisms and, we obtain that the sum + l in S l of all expressions of the form (a 1 ) (a i ) where k k i = l and k 1,, k i 0 is equal to the sum + l in S l of all expressions of the form (b 1 ) (b j ) where + + = l and,, 0, and that ψ 0 (a 1 ) ψ 0 (a i ) = ψ 0 (b 1 ) ψ 0 (b j ). These conditions, in turn, by definition of the (m + 1)-ary product in m (S m,, S 1, S 0 ), are equivalent to the desired equality

13 We end this section with a criterion to determine when an identity is satisfied in the (m + 1)-ary product m (J 1,, J 1, ). Corollary 7. Let A be a finite alphabet, m be a positive integer and be a tuple of positive integers. An identity u = v on A holds in m (J 1,..., J 1, ) if and only if u v. Consequently, an A- generated monoid S belongs to m (J 1,, J 1, ) if and only if S is a morphic image of A * /(m, ), or m(j 1,, J 1, ) =. Proof. Let S0 be the multiplicatively written monoid F A ( ) with unit 1. Let S i (1 i m) be the additively written commutative monoid (J 1 ) with operation + i and unit 0 i. Here B i denotes F A ( ) A (F A ( ) (A {1})) i-1 F A ( ). Consider for 0 i,j m with i + j m, the function h i,j : S i S j S i+j defined by for all x 0,,x i, y 0,,y j S 0, a 1, b 1 A, and a 2,,a i, b 2,, bj (A {1}), and the associated (m + 1)-ary product m (S m,,s 1, S 0 ). Now consider the one-to-one morphism of Theorem 6, where for all a A, ψ(a) = ((1, a, ),, (1, a, ), (1, a, 1), a). If u and v are words on A, then m (J 1,,J 1, ) = u = v if and only if F A ( m (J 1,, J 1, )) = u = v if and only if ψ(u) = ψ(v) if and only if u v. To see that ψ(u) = ψ(v) and u v are equivalent, the morphism ψ maps the word u = a 1 a i into an (m + 1)-tuple and ψ maps v = b 1 b j into an (m + 1)-tuple where u l, v l (1 l m) and u 0, v 0 are written in (3) and (4) respectively. As in the proof of Theorem 6, we suppose that i, j m; the other cases are simpler. The equality ψ(u) = ψ(v) holds if and only if corresponding components of the (m + 1)-tuples (5) and (6) are equal. The condition the lth components of (5) and (6) are equal is equivalent to S l = u l = v l or (u) = (v) for all 1 l m, and the condition the last components of (5) and (6) are equal is equivalent to S 0 = u 0 = v 0 or u v. The set of conditions (u) = (v),, (u) = (v) (which is clearly equivalent to (u) = (v) by Definition 2 implies the condition u v. Hence, by Lemma 1 we conclude that ψ(u) = ψ(v) if and only if u v. Corollary 8. We have V 2 = (J 1,,J 1, J 1 ). Proof. By Corollary 7 and the fact that V 2 = (m,1). References: [1] Almeida, J., Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994.

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